development

Let $X$ be a topological space. A development for $X$ is a countable collection $F_{1},F_{2},\\ldots$ of open coverings of $X$ such that for any closed subset $C$ of $X$ and any point $p$ in the complement of $C$ , there exists a cover $F_{j}$ such that no element of $F_{j}$ which contains $p$ intersects $C$ . A space with a development is called developable.

A development $F_{1},F_{2},\\ldots$ such that $F_{i}\\subset F_{i+1}$ for all $i$ is called a nested development. A theorem from Vickery states that every developable space in fact has a nested development.

References

1 Steen, Lynn Arthur and Seebach, J. Arthur, Counterexamples in Topology, Dover Books, 1995.

单词	Development
释义	development Let $X$ be a topological space. A development for $X$ is a countable collection $F_{1},F_{2},\\ldots$ of open coverings of $X$ such that for any closed subset $C$ of $X$ and any point $p$ in the complement of $C$ , there exists a cover $F_{j}$ such that no element of $F_{j}$ which contains $p$ intersects $C$ . A space with a development is called developable. A development $F_{1},F_{2},\\ldots$ such that $F_{i}\\subset F_{i+1}$ for all $i$ is called a nested development. A theorem from Vickery states that every developable space in fact has a nested development. References 1 Steen, Lynn Arthur and Seebach, J. Arthur, Counterexamples in Topology, Dover Books, 1995.
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